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Chapter 2: Problem 23

Solve each equation, and check the solution. $$ 2 m=15 $$

Short Answer

Expert verified
The solution to the equation is \(m = 7.5\).

Step by step solution

01

- Isolate the Variable

To solve the equation for \(m\), we need to isolate the variable \(m\) on one side of the equation. The equation given is \(2m=15\). Divide both sides of the equation by 2.
02

- Perform Division

Divide both sides of the equation \(2m = 15\) by 2 to isolate \(m\). So, \(m = \frac{15}{2} = 7.5\).
03

- Check the Solution

Substitute \(m\) back into the original equation to verify the solution. The original equation is \(2m = 15\). Substitute \(m = 7.5\): \(2 \times 7.5 = 15\). Since the left side equals the right side, the solution is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Isolating Variables
Isolating variables in algebra means getting the variable by itself on one side of the equation. When you face an equation like \(2m = 15\), your goal is to make it look like \(m = \text{some number}\). This step involves using inverse operations.

For example, if you see multiplication by 2 (like in our case, \(2m\)), you do the opposite operation (which is division by 2) on both sides of the equation. This step-by-step process helps to simplify the problem. Remember, whatever operation is done on one side must be done on the other to keep the equation balanced.
Division in Algebra
Division in algebra often helps in isolating the variable. For instance, to solve the equation \(2m = 15\), you divide both sides by 2, because 2 is what is multiplying \(m\).

Let's break it down:
  • Start with the equation: \(2m = 15\)
  • Divide both sides by 2: \(\frac{2m}{2} = \frac{15}{2}\)
The 2's cancel out on the left, leaving you with \(m = \frac{15}{2}\). Simplifying \(\frac{15}{2}\) gives you 7.5, so \(m = 7.5\).

This process shows how division is used to isolate the variable, solving the equation smoothly.
Checking Solutions in Algebra
It's important to check your solutions in algebra to verify their correctness. To check the solution, substitute the value of the variable back into the original equation.

For our equation \(2m = 15\), we found that \(m = 7.5\). Now, substitute 7.5 back into the original equation to check:
  • Original equation: \(2m = 15\)
  • Substitute \(m = 7.5\): \(2 \times 7.5\)
  • Calculate: \(2 \times 7.5 = 15\)
Since the left side equals the right side (both are 15), the solution \(m = 7.5\) is correct. This step ensures you didn鈥檛 make any mistakes along the way and that your solution is valid.

Always remember to verify your answers in algebra by substituting them back into the original equation!

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